(This is true whichever definition of average velocity is used.)
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On the subject of your question about the lab with the phase change (2.301 rad), I cannot understand either. Rotation of the axes of the ‘generating circle’ is equivalent to making any phase change you please.
My only comment is that the x component will not vary as sin(), but as cos(). The question doesn’t really make sense. The only way rotating the axes of the generating circle could not give you the new required curve is if the amplitude or the frequency of oscillation had changed – but from what you have said, that doesn’t appear to be the case. Sorry I can’t help more.